Dirichlet's ellipsoidal problem

In the winter of 1856–57, Dirichlet found some solutions of Euler equations and he presented those in his lectures on partial differential equations in July 1857 and published the results in the same month.[4] His work was left unfinished at his sudden death in 1859, but his notes were collated and published by Richard Dedekind posthumously in 1860.[5]

Bernhard Riemann said, "In his posthumous paper, edited for publication by Dedekind, Dirichlet has opened up, in a most remarkable way, an entirely new avenue for investigations on the motion of a self-gravitating homogeneous ellipsoid. The further development of his beautiful discovery has a particular interest to the mathematician even apart from its relevance to the forms of heavenly bodies which initially instigated these investigations."

Dirichlet's problem is generalized by Bernhard Riemann in 1860[6] and by Norman R. Lebovitz in modern form in 1965.[7] Let ${\displaystyle a_{1}(t),\ a_{2}(t),\ a_{3}(t)}$ be the semi-axes of the ellipsoid, which varies with time. Since the ellipsoid is homogeneous, the constancy of mass requires the constancy of the volume of the ellipsoid,

${\displaystyle a_{1}(t)a_{2}(t)a_{3}(t)=a_{1}(0)a_{2}(0)a_{3}(0)}$

same as the initial volume. Consider an inertial frame ${\displaystyle (X_{1},X_{2},X_{3})}$ and a rotating frame ${\displaystyle (x_{1},x_{2},x_{3})}$, with ${\displaystyle \mathbf {L} (t)}$ being the linear transformation such that ${\displaystyle \mathbf {x} =\mathbf {L} \mathbf {X} }$ and it is clear that ${\displaystyle \mathbf {L} }$ is orthogonal, i.e., ${\displaystyle \mathbf {L} \mathbf {L} ^{T}=\mathbf {L} ^{T}\mathbf {L} =\mathbf {I} }$. We can define an anti-symmetric matrix with this,

${\displaystyle \mathbf {\Omega } ^{*}={\frac {d\mathbf {L} }{dt}}\mathbf {L} ^{T}}$

where we can write the dual ${\displaystyle \mathbf {\Omega } }$ of ${\displaystyle \mathbf {\Omega } ^{*}}$ as ${\displaystyle \Omega _{ij}^{*}=\varepsilon _{ijk}\Omega _{k}}$ (and ${\displaystyle 2\Omega _{i}=\varepsilon _{ijk}\Omega _{jk}^{*}}$), where ${\displaystyle \mathbf {\Omega } (t)}$ is nothing but the time-dependent rotation of the rotating frame with respect to the inertial frame.

Without loss of generality, let us assume that the inertial frame and the moving frame coincide initially, i.e., ${\displaystyle \mathbf {X} (0)=\mathbf {x} (0)}$. By definition, Dirichlet's problem is looking for a solution which is a linear function of initial condition ${\displaystyle \mathbf {X} (0)=\mathbf {x} (0)}$. Let us assume the following form,

${\displaystyle X_{i}(t)=\sum _{j=1}^{3}P_{ij}(t){\frac {x_{j}(0)}{a_{j}(0)}}.}$

and we define a diagonal matrix ${\displaystyle \mathbf {A} (t)}$ with diagonal elements being the semi-axes of the ellipsoid, then above equation can be written in matrix form as

${\displaystyle \mathbf {X} =\mathbf {P} \mathbf {A} _{0}^{-1}\mathbf {x} (0)}$

where ${\displaystyle \mathbf {A} _{0}=\mathbf {A} (0)}$. It can shown then that the matrix ${\displaystyle \mathbf {S} =\mathbf {A} ^{-1}\mathbf {L} \mathbf {P} }$ transforms the vector ${\displaystyle \mathbf {A} _{0}^{-1}\mathbf {x} (0)}$ linearly to the same vector at any later time ${\displaystyle \mathbf {A} ^{-1}\mathbf {x} }$, i.e., ${\displaystyle \mathbf {A} ^{-1}\mathbf {x} =\mathbf {S} \mathbf {A} _{0}^{-1}\mathbf {x} (0)}$. From the definition of ${\displaystyle \mathbf {A} }$, we can realize the vector ${\displaystyle \mathbf {A} ^{-1}\mathbf {x} }$ represents a unit normal on the surface of the ellipsoid (true only at the boundary) since a fluid element on the surface moves with the surface. Therefore, we see that ${\displaystyle \mathbf {S} }$ transforms one unit vector on the boundary to another unit vector on the boundary, in other words, it is orthogonal, i.e., ${\displaystyle \mathbf {S} \mathbf {S} ^{T}=\mathbf {S} ^{T}\mathbf {S} =\mathbf {I} }$. In a similar manner as before, we can define another anti-symmetric matrix as

${\displaystyle \mathbf {\Lambda } ^{*}={\frac {d\mathbf {S} }{dt}}\mathbf {S} ^{T}}$,

where its dual is defined as ${\displaystyle \Lambda _{ij}^{*}=\varepsilon _{ijk}\Lambda _{k}}$ (and ${\displaystyle 2\Lambda _{i}=\varepsilon _{ijk}\Lambda _{jk}^{*}}$). The problem is one of uniform vorticity ${\displaystyle {\boldsymbol {\zeta }}}$ with components given by

${\displaystyle \zeta _{k}=-{\frac {a_{i}^{2}+a_{j}^{2}}{a_{i}a_{j}}}\Lambda _{k},\quad (i\neq j\neq k\neq i).}$

The pressure can take only quadratic form, can be seen from the momentum equation (and using the vanishing condition at the surface) given by

${\displaystyle p=p_{c}(t)\left(1-\sum _{i=1}^{3}{\frac {x_{i}^{2}}{a_{i}^{2}}}\right)}$

where ${\displaystyle p_{c}(t)}$ is the central pressure, so that ${\displaystyle \nabla p=-2p_{c}\mathbf {A} ^{-2}\mathbf {x} }$. Finally, the tensor momentum equation reduces to

${\displaystyle {\begin{aligned}&{\frac {d^{2}\mathbf {A} }{dt^{2}}}+{\frac {d}{dt}}(\mathbf {A} \mathbf {\Lambda } ^{*}-\mathbf {\Omega } ^{*}\mathbf {A} )+{\frac {d\mathbf {A} }{dt}}\mathbf {\Lambda } ^{*}-\mathbf {\Omega } ^{*}{\frac {d\mathbf {A} }{dt}}+\mathbf {A} \mathbf {\Lambda } ^{*2}+\mathbf {\Omega } ^{*2}\mathbf {A} -2\mathbf {\Omega } ^{*}\mathbf {A} \mathbf {\Lambda } ^{*}\\[8pt]={}&{-2}\pi G\rho \mathbf {B} \mathbf {A} +{\frac {2p_{c}}{\rho }}\mathbf {A} ^{-1}\end{aligned}}}$

where ${\displaystyle G}$ is the gravitational constant and ${\displaystyle \mathbf {B} }$ is diagonal matrix, whose diagonal elements are given by

${\displaystyle B_{i}=a_{1}a_{2}a_{3}\int _{0}^{\infty }{\frac {du}{(a_{i}^{2}+u){\sqrt {(a_{1}^{2}+u)(a_{2}^{2}+u)(a_{3}^{2}+u)}}}}.}$

The tensor momentum equation and the conservation of mass equation, i.e., ${\displaystyle a_{1}a_{2}a_{3}=a_{1}(0)a_{2}(0)a_{3}(0)}$ provides us with ten equations for the ten unknowns, ${\displaystyle a_{1},\ a_{2},\ a_{3},\ p_{c},\ \mathbf {\Lambda } ,\ \mathbf {\Omega } .}$

It states that if a motion determined by ${\displaystyle \mathbf {X} (t)=\mathbf {P} (t)\mathbf {A} _{0}^{-1}\mathbf {x} (0)}$ is admissible under the conditions of Dirichlet's problem, then the motion determined by the transpose ${\displaystyle \mathbf {P} ^{T}}$ of ${\displaystyle \mathbf {P} }$ is also admissible. In other words, the theorem can be stated as for any state of motions that preserves a ellipsoidal figure, there is an adjoint state of motions that preserves the same ellipsoidal figure.

By taking transpose of the tensor momentum equation, one sees that the role of ${\displaystyle \mathbf {\Lambda } ^{*}}$ and ${\displaystyle \mathbf {\Omega } ^{*}}$ are interchanged. If there is solution for ${\displaystyle \mathbf {A} ,\ \mathbf {\Lambda } ^{*},\ \mathbf {\Omega } ^{*}}$, then for the same ${\displaystyle \mathbf {A} }$, there exists another solution with the role of ${\displaystyle \mathbf {\Lambda } ^{*}}$ and ${\displaystyle \mathbf {\Omega } ^{*}}$ interchanged. But interchanging ${\displaystyle \mathbf {\Lambda } ^{*}}$ and ${\displaystyle \mathbf {\Omega } ^{*}}$ is equivalent to replacing ${\displaystyle \mathbf {P} }$ by ${\displaystyle \mathbf {P} ^{T}}$. The following relations confirms the previous statement.

${\displaystyle \mathbf {P} =\mathbf {L} ^{T}\mathbf {A} \mathbf {S} }$

where, further

${\displaystyle {\frac {d\mathbf {S} }{dt}}=\mathbf {\Lambda } ^{*}\mathbf {S} ,\quad {\frac {d\mathbf {L} }{dt}}=\mathbf {\Omega } ^{*}\mathbf {L} ,\quad {\text{and}}\quad \mathbf {S} (0)=\mathbf {L} (0)=\mathbf {I} .}$

The typical configuration of this theorem is the Jacobi ellipsoid and its adjoint is called as Dedekind ellipsoid, in other words, both ellipsoid have same shape, but their internal fluid motions are different.

• Maclaurin ellipsoid
• Jacobi ellipsoid